By Jerrold Franklin
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This result, which is essentially Eq. (33, can be easily obtained when the coupling constant is independent of the initial and final states of the electron by integrating the Fermi golden rule equation over the final energies of the electron [Eq. (29)]. Thus, the scattering rates of Shichijo and Hess are derived from perturbation theory. The application of the golden rule in this high-scattering-rate region is questionable, as pointed out by Capasso et al. ( 1981a) and extensively discussed by Barker ( 1973, 1980).
The probability P(Ei)is proportional to the energy-distribution function f(Ei),which in turn can be obtained by solving the Boltzmann equation for the momentum distribution function f(p). The Boltzmann equation, assuming time independence, a uniform field, and absence of diffusion effects (Chynoweth, 1968),becomes )>7 eF * V,f(P) = I dP’[f(P’)W(P’, PI - f ( P ) W p , P’)l, (41) which simply states that at steady state, the rate of change of the distribution due to acceleration by the field must balance the rate of change due to collisions; W(p, p ’) is the probability per unit time of a carrier with momentum p being scattered into the element of volume dp’ at p’.
This means that the distribution function is neither a spike in the direction of the field (Shockley’s ballistic electrons) nor the nearly isotropic distribution of Wale the two aspects are complementary. Baraff assumed a velocity distribution of the form j - ( ~cos , e) = A ( U ) + ~ ( vs(i) - cos e), (57) which is a mixture of a spherical part and a “spike” in the direction of the electric field. By inserting this expression in the Boltzmann equation [Eq. (41)], one can find the coefficients A ( v ) and B(v)that best represent the true distribution function.